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Originally Posted by: Ber7 Refined the algorithm and added examples
Hi. This solver is great, it's very stable and works in cases where I can't figure how it do things right. In the attached, I just clean the code a little, but the application it's entirely the same. DistMod 2.sm (48kb) downloaded 22 time(s). DistMod 2.pdf (168kb) downloaded 15 time(s).Best regards. Alvaro.

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Thank you very much, Alvaro, for modifying the program. In your version I reduced the number of arguments in the Dist and al_nleqsolve functions. Your program began to work much faster. DistRazonar.sm (56kb) downloaded 20 time(s).




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Hi Ber. Added Lagrange multipliers for refine the last nearest point in the curve. this enables to decrease the number of steps, up to a factor of ten. I guess that the al_nleqsolve calls could be implemented as Draghilev method too. DistBer.sm (54kb) downloaded 26 time(s). DistBer.pdf (182kb) downloaded 18 time(s).Best regards. Alvaro.

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Originally Posted by: Razonar I guess that the al_nleqsolve calls could be implemented as Draghilev method too. Is that what you mean ? Contour_10 Draghilev [Electron].sm (26kb) downloaded 26 time(s).




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Originally Posted by: Jean Giraud Originally Posted by: Razonar I guess that the al_nleqsolve calls could be implemented as Draghilev method too. Is that what you mean ? Contour_10 Draghilev [Electron].sm (26kb) downloaded 26 time(s). Hi Jean. Yes, that, and the use of Dragilev (without h, I forgot that) method for solve nonlinear equations, as uni teach sometime ago. Best regards. Alvaro.




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Originally Posted by: Razonar Hi Jean. Yes, that, and the use of Dragilev (without h, I forgot that) method for solve nonlinear equations, as uni teach sometime ago.
From recollection, original Uni [Viacheslav] further exploited by Ber7 [smart cat !] Buenos dias, Amigo.




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Finding Distance of a Point from SurfaceThe difference from the program distance to the curve in only one expression: in the top line instead of i: = (1 ... 2)) we have i: = (1 ... 3). A is a given point, a is its projection to the surface, Аa (k) is the length of the segment Аa (k) to the surface, the hstep of decreasing the distance, a (k) is the intermediate coordinate of the point touch.The red line is the normal, the green is the trajectory of the movement of the point a (k) from the initial the position of a (1) to the final a.I borrowed the trajectory demonstration from Razonar,to whom I express my gratitude. Dist3d.sm (64kb) downloaded 26 time(s).

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You may have interest in that stuff:
[url=https://www.geometrictools.com/Documentation/DocUpdateHistory.pdf]

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Originally Posted by: Ber7 Finding Distance of a Point from Surface The difference from the program distance to the curve in only one expression: in the top line instead of i: = (1 ... 2)) we have i: = (1 ... 3).
Hi. Attached version have some reformulations, but is essentially the same, just added Lagrange multipliers for refine last point and join 3D and 2D in only one procedure. Maybe don't works in all cases, I guess that because use less points, but is more pedagogical from my point of view. Dist.sm (173kb) downloaded 19 time(s). Dist.pdf (501kb) downloaded 19 time(s).Best regards. Alvaro.

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Thank you, Razonar, for improving the program. I would like to have a criterion to verify the accuracy of our calculations. For example, you can use the vector product of unit vectors dist and normals, whose modulus with perfect accuracy will be equal to to zero. he magnitude of the deviation from zero characterizes the accuracy. Agree Are you with this criterion? Cross.sm (66kb) downloaded 20 time(s).

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Originally Posted by: Ber7 Thank you, Razonar, for improving the program. I would like to have a criterion to verify the accuracy of our calculations. For example, you can use the vector product of unit vectors dist and normals, whose modulus with perfect accuracy will be equal to to zero. he magnitude of the deviation from zero characterizes the accuracy. Agree Are you with this criterion? Cross.sm (66kb) downloaded 20 time(s). Hi Ber. Thanks for say "our" calculations, but actually are yours. I just make some few modifications. Some general notes: It's a numerical procedure, so, guess values are part of the solution. Because of that, and the fact that you take a random external point I guess that my major modification can't get the global solution, and sometimes get only local ones. Each curve or surface need an specific general expression for the guess value in such case. This is a small modification of your implementation, with some notes: Cross.sm (74kb) downloaded 15 time(s).This is the modified implementation, and shows that the introduction of the Lagrangian multipliers for refine the last point get a very small cross third component value in all the cases where can find a solution. Dist.sm (191kb) downloaded 15 time(s).This shows the importance of the initial guess value for seek the global solution Dist_InitialGuess.sm (72kb) downloaded 17 time(s).So, answering your question, my conclusion is: Yes, you can use cross to show that, in 2D case the circle is locally tangent to the curve and, for the 3D case, a sphere is. But guess that for construction of your algorithm can't be used as a stop criteria because in your setup you need to move the point along the curve for seek the better approach. But what I see is that I can find only local solutions, but not the global one. For that, given a random external point A to a curve or a surface you must to get the initial guess xo as a function of the given function and the point A, that's specific for each function. Dragilev method can help here for finding the global distance because with that you know that you have a curve parametrization, so guess that only need to know in which direction you need to move, and estimate for how many 'time' you need to do that. Next, you use your algorithm and refine the last point with the Laplace multiplier to ensure that the circle centered in A is tangent to the curve. But for the 3D case I don't see any help here, because with two parameters you lost the sort property of reals: plane points can't be sorted. Hope that this notes helps. Best regards. Alvaro.

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“Finding Distance of a Point from curve or surface”  not far the best application of the Draghilev method. An example of the projection of points from one surface to another using the optimization methods in Maple. Edited by user 05 April 2019 15:29:57(UTC)
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Originally Posted by: алексей_алексей âFinding Distance of a Point from curve or surfaceâ If Draghilev method wouldn't have been discovered, it could be solved otherwise, by more conventional method ... like in the attached 1. Distance between user random triplets pairs. 2. Specified "user random" from one pair triplet [surface/solid] 3. Direction cosines plane [normal to solid] as one pair triplet the 2nd triplet pair from solid. Easy but not included: distance between helix/randomBrownian. I'm lost in that thread, further latest SS don't read previous stable SS. If Mathcad 11 and earlier can do w/o Draghilev, Smath earlier as well. Cheers ... Jean Vector Maths Direction Cosine [DistinSpace] Copy.sm (83kb) downloaded 19 time(s).




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It seems to me that the Draghilev method is intended for a “large” number of variables with a “small” number of equations. For example, as in the case of solving the inverse problem of kinematics for a threelink manipulator with a variable length of the working link. This manipulator has 6 degrees of freedom.




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Hi. The distance for 2D parametrizing plane curves by Dragilev method for seek a good guess for using with Laplace multipliers method for minimizing the distance. Dist2Dragilev.sm (72kb) downloaded 26 time(s). Dist2Dragilev.pdf (375kb) downloaded 22 time(s).Best regards. Alvaro.

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Point Orthogonal Projection onto a flat or spatial Curve
Calculation procedure 1. We will use Draghilev's method a) to parameterizethe curves and find a good initial guess. We will use the point a0, found by the minimum distance from the curve to a given point P. b) to calculate the unit vector of the tangent T (x) to a point on the curve. 2 To the system of equations we add an equation that expresses the conditions of perpendicularityof the tangent at the point x of the curve and the segment connecting the points P and x. This means that the inner product of the unit tangent vector to the curve f (x) and the unit vector Px is equal to zero. As a result, we obtain the system of equations 3. We solve the system with the initial assumption a0 distantion.sm (125kb) downloaded 8 time(s). Trascidental curves.pdf (395kb) downloaded 11 time(s).Edited by moderator 02 October 2020 22:32:24(UTC)
 Reason: Fixed bbcode

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Draghilev's method and an approach to modeling platforms (parallel manipulators) with the number of degrees of freedom from 1 to 6.Forum MaplePrimes 1 and 6Edited by user 20 November 2020 22:36:31(UTC)
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